Mathematical Methods for MBS in Descriptor Form

Abstract

Recall from section 1.3 that we want to consider IVP for the descriptor form of the equations of motion of MBS, where we assume that an index reduction to index one has been performed, e.g., by employing a multibody formalism like MBSNAT. This means that we deal with a semi-explicit DAE system with linearly-implicit algebraic equations in the variables p, v, a and λ: $\${\textbackslash}begin{array}{*{20}{c}} \{{\textbackslash}dot p = v}&;&\{{\textbackslash}dot v = a} {\textbackslash}end{array}$$ (3.0.1a)$\${\textbackslash}left( \{{\textbackslash}begin{array}{*{20}{c}} {M(p)}&{G{{(p)\}{\textasciicircum}T}} {\textbackslash}{\textbackslash} {G(p)}&0 {\textbackslash}end{array}} {\textbackslash}right){\textbackslash}left( \{{\textbackslash}begin{array}{*{20}{c}} a {\textbackslash}{\textbackslash} {\textbackslash}lambda {\textbackslash}end{array}} {\textbackslash}right) = {\textbackslash}left( \{{\textbackslash}begin{array}{*{20}{c}} {f(t,p,v)} {\textbackslash}{\textbackslash} \{{\textbackslash}gamma (p,v)} {\textbackslash}end{array}} {\textbackslash}right)$$ (3.0.1b) subject to the constraints $\${\textbackslash}begin{array}{*{20}{c}} {g(p) = 0}&{(position constraint)} {\textbackslash}end{array}$$ (3.0.1c)$\${\textbackslash}begin{array}{*{20}{c}} {G(p)v = 0}&{(velocity constraint)} {\textbackslash}end{array}$$ (3.0.1d) with the consistent initial values $${p_0}: = p({t_0}),{v_0}: = v({t_0}),{a_0}: = a({t_0}), and \{{\textbackslash}lambda _0}: = {\textbackslash}lambda ({t_0}).$$ (3.0.1e)